Studies in Rings generalised Unique Factorisation Rings

-83-
Theorem 3.25
If A and B are two ideals of a ring R and
C>= {P 1
... pJ is a collection of prime ideals of R
t
with A~B ~ U p . , then eith er A ~ B or A s P. for
1 1
i=l
some i.
Theorem 3.26
Let D and 6' be finite coLl.e c t i.o n s of non zero
prime ideals in a ring R with neither P ~ Q nor Q ~ p
for any PGA and Q € ~l Then there exist at least one
element u E
Proof:
n
PG~
P such tha t u t U Q.
Q E 6'
Let 6 = {:1 ••• pJ and6'= [Q 1
••• Qm}· Then
m
U
i=l Q.•
1
m
For, if PI 5 U Q. , < m U Q.•
i=l
1 i=l
1
Thus by "pr i me avoidance" ei th er P l
= 0 or PI < Q. for
J
some j , which is impossible. Similarly
m
m
P2 $ U Q.• Denote V Q. = u. Then there exists
1
1
i=l
i=l
01= P 1
€' P 1
such that P1~ U and 01= P2 E P 2
such that
P2 t U. Now P1RP2 1= O. For, if P1RP2 = 0, then